Critical behavior of the percolation transition and low-temperature corrections in the Abrikosov-Moukhin short-range damped-RKKY spin glass

In a metallic spin glass whose conduction-electron mean free path $l$ is shortened by many nonmagnetic defects so that $n_m^{1/3}l\ll1$, the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange between magnetic impurities becomes exponentially damped and effectively short ranged, and each impurity couples dominantly to its single nearest neighbor. Abrikosov and Moukhin (1978) built a microscopic theory of this regime - specific heat, susceptibility, and resistivity across five temperature ranges, with a percolation-type transition at $\beta=\beta_c\approx3$ where an infinite cluster of rigidly bound spins forms - but left the critical region and the corrections of that theory undetermined. This problem asks for its completion: a microscopic determination of the exponent $a$ in the conjectured singular specific-heat scaling $C_{\mathrm{sing}}\sim n_m(\beta_c-\beta)^a$ (or a demonstration that no singular part exists), the quantitative susceptibility critical behavior near the transition for the true three-dimensional positional percolation problem with oscillatory damped random-sign bonds (beyond the Bethe-lattice approximation used in the source), the quantitative modification of the low-temperature asymptotics by spin-lattice and dipole-dipole interactions (which cut off the asymptotics at $T_d\sim\mu^2(4\pi n_m l)^{3/2}$), and the material-specific modeling required to identify an experimental realization.

Background

Metallic spin glasses and the RKKY interaction. A spin glass is a dilute magnetic alloy - the canonical examples are CuMn and AuFe - in which magnetic impurities sit at random substitutional positions and, below a freezing temperature, enter a state with no spatial periodicity but with a reproducible, history-dependent magnetic response. Two impurities are coupled by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, the indirect exchange mediated by conduction electrons: it oscillates in sign with the inter-impurity separation $r$ as $\cos(2p_0r)$ and decays as $r^{-3}$, where $p_0$ is the electron Fermi momentum. Random positions times an oscillating sign produce frustration - not all bonds on a cluster can be satisfied simultaneously - which is the microscopic origin of glassy freezing. Experimentally the susceptibility shows a cusp at the freezing temperature while the specific heat shows only a broad, weak maximum; whether any thermodynamic singularity accompanies the cusp has been a recurring question of the field.

The short-range, many-defect regime. In a dirty metal with a high concentration of nonmagnetic defects, the conduction-electron mean free path $l$ (the average distance an electron travels between elastic scattering events) is short, and since the conduction electrons mediate the coupling, the RKKY interaction acquires an exponential damping factor $e^{-r/l}$. Abrikosov and Moukhin considered the regime $n_m^{1/3}l\ll1$ (equivalently $4\pi n_m l^3\ll1$), where $n_m$ is the magnetic-impurity concentration and $r_0\sim n_m^{-1/3}$ the typical inter-impurity spacing: the typical impurity then has effectively only one interaction partner, its nearest neighbor, and the pair interaction is

$$H_{12}=-V_0\,(\mathbf{S}_1\cdot\mathbf{S}_2)\,\frac{\cos(2p_0 r)}{r^3}\,e^{-r/l},\qquad V_0=\left(\frac{J}{n}\right)^2\frac{p_0 m}{4\pi^3},$$

with $J$ the exchange constant between conduction electrons and an impurity, $n$ the atomic density of the host, and $m$ the electron mass. Treating the impurity positions as a Poisson gas and the oscillation factor $q=|\cos(2p_0r)|$ as distributed with weight $(2/\pi)(1-q^2)^{-1/2}dq$, they obtained the specific heat, susceptibility, and resistivity by averaging single-impurity and pair contributions. Five temperature regimes appear, set by the scales $T_0$, the transition temperature $\theta$, and $T_1=V_0S^2/l^3$: a very-low-temperature regime $T\ll T_0$ dominated by rare, weakly bound single spins; an intermediate regime $T_0\ll T\ll\theta$ dominated by bonds of energy of order $T$ at logarithmically large separations $r\approx l\ln(V_0S^2n_m/T)$; the transition region near $\theta$; a virial-type regime $\theta\ll T\ll T_1$ (virial: a cluster expansion in the impurity concentration in which single spins and bound pairs are counted and larger clusters are neglected); and the high-temperature regime $T\gg T_1$ where the standard undamped RKKY virial results are recovered.

The percolation picture of the transition. Percolation is the geometric transition at which, as the density of randomly placed connected objects grows, a cluster spanning the whole sample first appears. The source defines a temperature-dependent interaction radius $r(T)$ by equating the bond energy to the temperature, $S^2V_0e^{-r(T)/l}r(T)^{-3}=T$, and a percolation parameter $\beta=(4\pi/3)n_m r^3(T)$, the mean number of impurities inside the interaction sphere. When $\beta$ reaches the continuum percolation threshold $\beta_c\approx3$ (a value taken from percolation studies of hopping conductivity), an infinite cluster of rigidly exchange-bound spins forms; the temperature at which this happens is the transition temperature $\theta$. Spins absorbed into the infinite cluster no longer follow a weak probing field, which produces the cusp in the susceptibility,

$$\chi(T<\theta)=\frac{\mathcal{N}\mu^2S^2}{3T}\,[1-P(\beta)],$$

where $\mu$ is the impurity moment, $\mathcal{N}$ the number of impurities, and $P(\beta)$ the percolation probability, which near threshold behaves as $(\beta-\beta_c)^{\alpha}$ with $\alpha\approx0.35$ as quoted in the source (the modern three-dimensional percolation order-parameter exponent is $\approx0.42$). Energy-related quantities - the specific heat and the resistivity - are argued to show essentially no singularity at $T=\theta$, their leading behavior being set by bonds of energy of order $T$ and by single-particle contributions. The singular part of the specific heat, if any, was only conjectured to scale as

$$C_{\mathrm{sing}}\sim n_m(\beta_c-\beta)^a$$

with an undetermined positive exponent $a$, parametrically small compared to the analytic background.

What the source calculation did not determine. (i) The exponent $a$ was not derived microscopically; its relation, if any, to the standard exponents of three-dimensional continuum percolation (order-parameter exponent $\beta_p\approx0.42$, correlation-length exponent $\nu\approx0.88$) was left conjectural. (ii) The susceptibility near $\theta$ was obtained by replacing the true three-dimensional positional percolation problem - Poisson-distributed impurity positions with oscillatory, exponentially damped, random-sign bonds - by a Bethe lattice (a Cayley tree, the infinite loopless branching graph on which percolation statistics are exactly solvable in a mean-field-like fashion) and by borrowing the percolation probability from standard percolation; the quantitative critical behavior of $\chi$ for the true problem, including the role of loops and the cluster statistics of the real geometry, was not computed. (iii) Spin-lattice anisotropy (the anisotropic coupling of a spin to the host lattice induced by the spin-orbit interaction) and magnetic dipole-dipole interactions were neglected. Dipolar couplings fall as $\mu^2/r^3$ and are not damped by disorder, so at the bond length $r_1=(4\pi n_m l)^{-1/2}$ that dominates the very-low-temperature asymptotics they impose a cutoff temperature $T_d\sim\mu^2(4\pi n_m l)^{3/2}$; the exponentially small crossover scale $T_0=V_0S^2n_m(4\pi n_m l^3)^{3/4}\exp[-(4\pi n_m l^3)^{-1/2}]$ must exceed $T_d$ for that regime to exist at all, and how the dipolar and spin-lattice terms quantitatively modify the low-temperature asymptotics and the cluster dynamics was left unresolved. (iv) The conditions for experimental realization were only sketched as inequalities: the Kondo temperature $T_K$ (below which conduction electrons screen the impurity moments and remove them from the magnetic problem - the Kondo effect) must satisfy $T_K\ll\theta$; the nonmagnetic-defect atomic concentration $c_i$ must satisfy $c_i\gg c_m^{1/3}$, where $c_m$ is the magnetic-impurity atomic concentration; and dipolar energies must stay below exchange energies. With optimistic parameters the source estimated $\theta\lesssim1$ K, pointing to amorphous metals with very dilute magnetic impurities, but no concrete material was identified and no measurable magnitudes were predicted.

Historical position. After 1978 the theory of spin glasses consolidated around the Edwards-Anderson model (spins on a lattice with quenched random nearest-neighbor couplings) and the Sherrington-Kirkpatrick model (its infinite-range mean-field version), whose glassy solution involves replica symmetry breaking (a formalism that averages over copies of the disordered system to describe the many-valley structure of the frozen phase), not around this microscopic damped-RKKY construction, although a modern line of work on disorder-damped RKKY interactions in dilute magnetic semiconductors, Kondo alloys, and disordered conductors has revived the underlying physical setting. The four questions above were raised by the source itself and have not been answered by the subsequent literature.

Problem Statement

Consider a metallic host containing a dilute concentration $n_m$ of magnetic impurities of spin $S$ and moment $\mu$ at Poisson-random positions, with enough nonmagnetic defects that the conduction-electron mean free path $l$ satisfies $n_m^{1/3}l\ll1$ (equivalently $4\pi n_m l^3\ll1$); impurity pairs interact via the damped oscillatory RKKY exchange (the indirect spin-spin exchange mediated by the conduction electrons) $H_{12}=-V_0(\mathbf{S}_1\cdot\mathbf{S}_2)\cos(2p_0r)\,r^{-3}e^{-r/l}$ with $V_0=(J/n)^2p_0m/4\pi^3$, where $p_0$ is the conduction-electron Fermi momentum, $J$ the exchange constant between conduction electrons and an impurity, $n$ the atomic density of the host, and $m$ the electron mass. Define the interaction radius $r(T)$ by $S^2V_0e^{-r(T)/l}r(T)^{-3}=T$, the percolation parameter $\beta=(4\pi/3)n_mr^3(T)$ (the mean number of impurities inside the interaction sphere) with threshold $\beta_c\approx3$, and the transition temperature $\theta$ by $\beta(\theta)=\beta_c$. The problem is to complete the microscopic theory of this model's critical region and low-temperature asymptotics - the theory whose non-critical regimes the source derived - by resolving the following four linked components that the source left open:

(1) Specific-heat singularity at the percolation threshold. Determine microscopically the exponent $a$ in the conjectured scaling $C_{\mathrm{sing}}\sim n_m(\beta_c-\beta)^a$ of the singular part of the specific heat as $\beta\to\beta_c$ (equivalently as $T\to\theta$), including its relation, if any, to the standard exponents of three-dimensional continuum percolation - or demonstrate that no singular part exists beyond the analytic background.

(2) Susceptibility critical behavior for the true positional percolation problem. Compute the quantitative critical behavior of $\chi(T)$ near $\theta$ for the actual three-dimensional positional percolation problem of Poisson-distributed impurities with oscillatory, exponentially damped, random-sign bonds, going beyond the Bethe-lattice (Cayley-tree) substitution and the borrowed percolation-probability form $P(\beta)\sim(\beta-\beta_c)^{\alpha}$ with $\alpha\approx0.35$ used in the source - in particular the shape of the cusp and the role of loops and of the cluster statistics of the true geometry.

(3) Interaction corrections to the low-temperature asymptotics. Quantify how spin-lattice anisotropy and magnetic dipole-dipole interactions modify the very-low-temperature ($T\ll T_0$) specific-heat, susceptibility, and resistivity $\rho$ asymptotics (where $\rho$ denotes the magnetic-impurity contribution to the electrical resistivity arising from exchange scattering of conduction electrons off the impurity spins, the quantity the source derived in the non-critical regimes) and the cluster dynamics near the transition; in particular decide first whether the consistency condition $T_0\gg T_d$, under which the exchange-dominated regime $T\ll T_0$ exists at all, holds anywhere in the model's applicability domain - a demonstration that no such window survives is itself a complete answer to this component - and, only if it does, determine the corrected asymptotics and the actual observable window in that regime, given the dipolar cutoff temperature $T_d\sim\mu^2(4\pi n_m l)^{3/2}$, since $T_0$ is exponentially small in $(4\pi n_m l^3)^{-1/2}$ while $T_d$ is only algebraically small - where $T_0=V_0S^2n_m(4\pi n_m l^3)^{3/4}\exp[-(4\pi n_m l^3)^{-1/2}]$.

(4) Experimental realization. Using material-specific parameters (pseudopotential strengths - the effective electron-ion scattering potentials of the host, which set the exchange $J$ - the distribution of nonmagnetic-defect scattering cross sections that sets $l$, impurity moments, and Kondo screening), identify concrete candidate systems satisfying the applicability conditions $n_m^{1/3}l\ll1$, $T_K\ll\theta$ (where $T_K$ is the Kondo temperature, below which conduction electrons screen the impurity moments and remove them from the magnetic problem), and dipolar smallness, and predict the measurable magnitudes of $C$, $\chi$, $\rho$, and $\theta$.

Components (1)-(3) are the critical-region and correction structure of one and the same microscopic model, and (4) is the experimental counterpart of that model. The four components are independent, jointly posed parts of the single question.

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